Superstring amplitudes, describing the scattering of superstrings, have perturbative power series expansions in the coupling constants, whose coefficients are integrals over moduli (super) spaces of supercurves. These integrals over non-compact superspaces are expected to (conditionally) converge, in contrast with the case of Feynmann diagrams of Quantum Field Theory, where integrals diverge and require renormalization. In this talk I will review the definition of these objects, present the challenges for a mathematical description (compactification, integration over middle dimensional cycles, GSO projection, dependence on the regularization), and explain how to address them in the simplest non-trivial case of the genus 2 contribution to the vacuum amplitude of the type II superstring in 10 dimensional Minkowski space. The talk is based on joint work with David Kazhdan and Alexander Polishchuck.